The present invention relates in general to well-logging operations, and is particularly useful in anisotropic formations wherein electromagnetic induction measurements are sensitive to horizontal and vertical properties of the formation.
In oil and gas exploration, measurement-while-drilling or wireline systems are utilized to develop logs of the earth formation surrounding the borehole. In general, electrical logging apparatus includes: a transmission system that energizes the earth formations with either an electromagnetic field or currents, and at least one receiver system for monitoring the effect that the earth formations (and borehole) have on the field or current.
The electrical properties of the earth formation provide information about the geologic materials that makeup the formations, and about their likely oil, gas, and water content. In addition, the dielectric properties of the earth formations are also of interest.
Some earth formations are isotropic. In the present context, this means that the effect that the formation has on an incident electromagnetic wave is independent of the direction of propagation of the incident wave. Other formations are anisotropic, which means that the earth formation may have a greater conductivity or dielectric constant in one direction than in another. This is particularly true of many sedimentary geologic formations in which the current flows more readily in a direction parallel to the bedding plane than in other directions due to the fact that a number of mineral crystals are flat or elongated, and they preferentially align with their long direction parallel to the sedimentary bedding plane at the time of deposition.
In oil and gas exploration, the anisotropy of earth formations can be expressed as the ratio of the horizontal and vertical property of interest. A commonly used model for the subsurface assumes that the formation is transversely isotropic ("TI"), characterized by a single axis of infinite symmetry. In such a medium, the properties in the direction of the symmetry axis are different from the properties in any direction perpendicular to the symmetry axis. It is assumed here that the axis of symmetry is vertical and perpendicular to the bedding plane of the formation. Cases where this assumption is not true can be handled in ways that would be familiar to those knowledgeable in the art.
For electrical properties of subsurface formations, a commonly used anisotropy factor is ##EQU1## where .sigma..sup.h and .sigma..sub.v are the horizontal and vertical conductivities of the formation. In situations where the borehole intersects the formations substantially perpendicular to the bedding planes, conventional induction and propagation well logging tools are sensitive almost exclusively to the horizontal component of the formation resistivity.
The determination of horizontal and vertical conductivities is complicated in directional drilling wherein the boreholes are inclined to the bedding plane. In such a case, the tool readings contain an influence from the vertical as well as the horizontal conductivity. The inclination of the borehole axis to the normal to the bedding plane is, in general, not known precisely.
Moran discusses a method for modeling the effect of formation anisotropy on resistivity logging measurements and shows that, in principle, values of horizontal and vertical conductivities can be derived from the measured values of the amplitude and phase of the induction-logging conductivity signal.
Rosthal (U.S. Pat. No. 5,329,448) discloses a method for determining the horizontal and vertical conductivities from a propagation or induction well logging device. The method assumes that .theta., the angle between the borehole axis and the normal to the bedding plane, is known. Conductivity estimates are obtained by two methods. The first method measures the attenuation of the amplitude of the received signal between two receivers and derives a first estimate of conductivity from this attenuation. The second method measures the phase difference between the received signals at two receivers and derives a second estimate of conductivity from this phase shift. Two estimates are used to give the starting estimate of a conductivity model and based on this model, an attenuation and a phase shift for the two receivers are calculated. An iterative scheme is then used to update the initial conductivity model until a good match is obtained between the model output and the actual measured attenuation and phase shift.
The relevant wave-number that governs the propagation of electromagnetic waves is given by an expression of the form EQU k.sup.2 =.epsilon..mu..omega..sup.2 +i.sigma..mu..omega. (2)
where k is the wave number, .epsilon. is the dielectric constant, .mu. is the magnetic permeability, .sigma. is the conductivity and .omega. the angular frequency. In an anisotropic medium, .epsilon., .mu. and .sigma. are anisotropic quantities.
Hagiwara shows that the log response of an induction-type logging tool can be described by an equation of the form ##EQU2## In equation 3, V is the measured signal at a distance L from the source and .beta. is given by EQU .beta..sup.2 =cos.sup.2 .theta.+.lambda..sup.2 sin.sup.2 .theta.(4)
and k is related only to the horizontal formation parameters.
Equation 3 is actually a pair of equations, one corresponding to the real part and one corresponding to the imaginary part of the measured signal, and has two unknowns. By making two measurements of the measured signal, the parameters k and .beta. can be determined. The two needed measurements can be obtained from (1) R and X signals from induction logs, (2) phase and attenuation measurements from induction tools, (3) phase or attenuation measurements from induction tools with two different spacings, or (4) resistivity measurements at two different frequencies. In the low frequency limit, .epsilon. can be neglected in equation 2 and from known values of .omega. and .mu., the conductivity .sigma. can be determined from k, assuming a value of .mu. equal to the permittivity of free space
Equation 2 shows that the dielectric effect can be quite large at high frequencies. In relatively high resistivity (low conductivity) formations, like oil bearing sands, it can be seen from equation 2 that the impact of the dielectric effect can be quite large even at relatively low frequencies. The prior art methods do not take into account the effect of dielectric properties of the formation.
Those knowledgeable in the art would recognize that equation 3 is a nonlinear equation and may have more than one solution. The prior art methods do not address this possibility of nonuniqueness of solutions to the equation.
The hardware devices used in prior art typically make measurements of amplitude attenuation and phase differences. These methods cannot be made accurately. As would be known to those familiar with the art, these errors in measurement result in corresponding errors in the estimates of resistivity. Prior art methods are deficient in that they do not give any estimate of the reliability of the estimated resistivity.
There is a need for an invention that gives estimates of the anisotropic resistivity parameters of an underground formation while properly accounting for the effect of the dielectric constant of the formation on propagating electromagnetic waves. Such an invention should also provide estimates of the uncertainty in the estimated resistivity parameters in terms of uncertainties in the measurements made by the device. It is desirable that the invention make redundant measurements to reduce the uncertainties in the estimation of the formation parameters. It is also desirable that the invention take into account the nonuniqueness of possible solutions to the problem of determining the formation parameters from measured values of the propagating electromagnetic signal. The present invention satisfies these needs.